Find The Value Of $x$ If $49^{x+1} + 7^{2x} = 350$.

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving a specific exponential equation, $49^{x+1} + 7^{2x} = 350$, and provide a step-by-step guide to finding the value of x.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be challenging to solve. However, with the right approach and techniques, we can simplify these equations and find the value of the variable. In this case, we have an equation with two exponential terms, $49^{x+1}$ and $7^{2x}$, and we need to find the value of x that satisfies the equation.

Simplifying the Equation

To simplify the equation, we can start by expressing the bases as powers of a common base. In this case, we can rewrite $49^{x+1}$ as $(72){x+1}$, which simplifies to $7^{2(x+1)}$. This gives us:

72(x+1)+72x=3507^{2(x+1)} + 7^{2x} = 350

Using Properties of Exponents

Now that we have simplified the equation, we can use the properties of exponents to further simplify it. Specifically, we can use the property that $a^{m+n} = a^m \cdot a^n$ to rewrite the first term:

72(x+1)=72x+2=72x⋅727^{2(x+1)} = 7^{2x+2} = 7^{2x} \cdot 7^2

Substituting this back into the equation, we get:

72x⋅72+72x=3507^{2x} \cdot 7^2 + 7^{2x} = 350

Factoring Out the Common Term

Now that we have simplified the equation, we can factor out the common term, $7^{2x}$:

72x(72+1)=3507^{2x} (7^2 + 1) = 350

Solving for x

Now that we have factored out the common term, we can solve for x. We can start by dividing both sides of the equation by $7^{2x}$:

72+1=35072x7^2 + 1 = \frac{350}{7^{2x}}

Using Logarithms

To solve for x, we can use logarithms. Specifically, we can take the logarithm of both sides of the equation:

log⁡(72+1)=log⁡(35072x)\log(7^2 + 1) = \log\left(\frac{350}{7^{2x}}\right)

Applying Logarithmic Properties

Now that we have taken the logarithm of both sides of the equation, we can apply logarithmic properties to simplify it. Specifically, we can use the property that $\log(a/b) = \log(a) - \log(b)$ to rewrite the right-hand side:

log⁡(72+1)=log⁡(350)−2xlog⁡(7)\log(7^2 + 1) = \log(350) - 2x \log(7)

Solving for x

Now that we have simplified the equation, we can solve for x. We can start by isolating the term with x:

2xlog⁡(7)=log⁡(350)−log⁡(72+1)2x \log(7) = \log(350) - \log(7^2 + 1)

Dividing Both Sides by 2log(7)

To solve for x, we can divide both sides of the equation by $2\log(7)$:

x=log⁡(350)−log⁡(72+1)2log⁡(7)x = \frac{\log(350) - \log(7^2 + 1)}{2\log(7)}

Evaluating the Expression

Now that we have solved for x, we can evaluate the expression:

x=log⁡(350)−log⁡(50)2log⁡(7)x = \frac{\log(350) - \log(50)}{2\log(7)}

Simplifying the Expression

To simplify the expression, we can use the property that $\log(a/b) = \log(a) - \log(b)$:

x=log⁡(7)2log⁡(7)x = \frac{\log(7)}{2\log(7)}

Cancelling Out the Common Term

Now that we have simplified the expression, we can cancel out the common term, $\log(7)$:

x=12x = \frac{1}{2}

The final answer is 0.5\boxed{0.5}.

Introduction

In our previous article, we explored the process of solving exponential equations, specifically the equation $49^{x+1} + 7^{2x} = 350$. We broke down the solution into manageable steps, using properties of exponents and logarithms to simplify the equation and find the value of x. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. It is a type of equation that can be challenging to solve, but with the right approach and techniques, we can simplify these equations and find the value of the variable.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, we can start by expressing the bases as powers of a common base. We can also use properties of exponents, such as the property that $a^{m+n} = a^m \cdot a^n$, to rewrite the terms and simplify the equation.

Q: What is the difference between a logarithmic and exponential equation?

A: A logarithmic equation is an equation that involves a variable in the logarithm, while an exponential equation is an equation that involves a variable in the exponent. Logarithmic equations can be used to solve exponential equations, and vice versa.

Q: How do I use logarithms to solve an exponential equation?

A: To use logarithms to solve an exponential equation, we can take the logarithm of both sides of the equation and apply logarithmic properties to simplify it. We can then isolate the term with the variable and solve for it.

Q: What is the property of logarithms that allows us to rewrite the equation?

A: The property of logarithms that allows us to rewrite the equation is the property that $\log(a/b) = \log(a) - \log(b)$. This property allows us to simplify the equation and isolate the term with the variable.

Q: How do I evaluate the expression and find the value of x?

A: To evaluate the expression and find the value of x, we can use the properties of logarithms and exponents to simplify the equation and isolate the term with the variable. We can then solve for x by dividing both sides of the equation by the coefficient of x.

Q: What is the final answer to the equation $49^{x+1} + 7^{2x} = 350$?

A: The final answer to the equation $49^{x+1} + 7^{2x} = 350$ is x = 0.5.

Q: Can I use this method to solve other exponential equations?

A: Yes, you can use this method to solve other exponential equations. The key is to simplify the equation using properties of exponents and logarithms, and then isolate the term with the variable to solve for it.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

  • Not simplifying the equation using properties of exponents and logarithms
  • Not isolating the term with the variable
  • Not using the correct properties of logarithms
  • Not evaluating the expression correctly

Conclusion

Solving exponential equations can be challenging, but with the right approach and techniques, we can simplify these equations and find the value of the variable. By understanding the properties of exponents and logarithms, and using the correct methods to solve the equation, we can find the value of x and solve the equation.